# Spin-$S$ designer hamiltonians and the square lattice $S=1$ Haldane nematic

## TL;DR

This paper develops a method to construct spin-$S$ lattice models that are both symmetric and efficiently simulatable, and demonstrates a new phase called Haldane nematic in an $S=1$ square lattice model.

## Contribution

It introduces a general strategy for creating Marshall positive, symmetric spin-$S$ Hamiltonians and applies it to realize a novel Haldane nematic phase in an $S=1$ model.

## Key findings

- Discovery of a Haldane nematic phase breaking lattice rotational symmetry
- First-order transition between Néel and Haldane nematic phases
- Efficient simulation of spin-$S$ models with arbitrary spin using world line Monte Carlo

## Abstract

We introduce a strategy to write down lattice models of spin rotational symmetric Hamiltonians with arbitrary spin-$S$ that are Marshall positive and can be simulated efficiently using world line Monte Carlo methods. As an application of our approach we consider a square lattice $S=1$ model for which we design a $3\times 3$ - spin plaquette interaction. By numerical simulations we establish that our model realizes a novel "Haldane nematic" phase that breaks lattice rotational symmetry by the spontaneous formation of Haldane chains, while preserving spin rotations, time reversal and lattice translations. By supplementing our model with a two-spin Heisenberg interaction, we present a study of the transition between N\'eel and Haldane nematic phase, which we find to be of first order.